Systems driven by $α$-stable noises could be very different from their Gaussian counterparts. Stationary states in single-well potentials can be multimodal. Moreover, a potential well needs to be steep enough in order to produce stationary states. Here, it is demonstrated that 2D systems driven by bi-variate $α$-stable noises are even more surprising than their 1D analogs. In 2D systems, intriguing properties of stationary states originate not only due to heavy tails of noise pulses, which are distributed according to $α$-stable densities, but also because of properties of spectral measures. Consequently, 2D systems are described by a whole family of Langevin and fractional diffusion equations. Solutions of these equations bear some common properties but also can be very different. It is demonstrated that also for 2D systems potential wells need to be steep enough in order to produce bounded states. Moreover, stationary states can have local minima at the origin. The shape of stationary states reflects symmetries of the underlying noise, i.e. its spectral measure. Finally, marginal densities in power-law potentials also have power-law asymptotics.
Ornstein-Uhlenbeck Process Driven by Multiple Dichotomous Noises
Sorbonne Université（索邦大学）；CNRS, Laboratoire de Physique Théorique de la Matière Condensée (UMR CNRS 7600)（法国国家科学研究中心，凝聚态物质理论物理实验室）；Dipartimento di Fisica, Sapienza Università di Roma（罗马智慧大学物理系）；Nanotech-CNR, UOS di Roma and INFN, Sezione di Roma（纳米技术-意大利国家研究委员会罗马单元及罗马分部INFN）；Department of Physics, Pohang University of Science and Technology (POSTECH)（浦项科技大学物理系）；Asia Pacific Center for Theoretical Physics (APCTP)（亚太理论物理中心）
